How does neural network classifier classify from just drawing a decision plane?
If you are working with supervised learning, each training example has a label. That label is your classification of the provided input. Just like linear or logistic regression, if your problem only has 2 classes (e.g. determine whether a tumor is malignant or not), your network will have a single output. An output value of 1.0 could represent one class and an output of 0.0 represents the other. This is basically determining which side of multi-dimensional plane your input is on. 

If you have more than 2 (m) classes, then you will have m outputs from your network. Each output will represent the likelihood of the input matching that class. For example if you have an image recognition network that has 4 classes: Dog, truck, boat, person, you would have 4 outputs with each one representing how likely that image is one of the classes. Think about this as 4 independent planes, where each one is making a determination: does this image match this class(1 of {dog, truck, boat, person}) or not. If you had outputs of dog: 0.3, truck: 0.25, boat: 0.7, person: 0.4, One simple algorithm would be simply pick the class with the maximum value -- in this case, it would classify this image as a boat.